Maths IA · Modelling · topic 34 of 40

Loans and savings: when does a plan break even? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle start, but compare plans!

Two loan offers, same amount, different small print. Which is cheaper, and when does the answer flip? You're going to prove it with algebra, not just a calculator.

Difficulty●○○ Gentle
KitA spreadsheet, real loan or savings offers, your GDC
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like money questions with a clear answer
  • ✓ you can compare real offers
  • ✓ you enjoy deriving a formula and then checking it

The big idea

Compound interest turns a loan or a savings plan into a geometric series. After n months the balance is the loan grown by interest, minus every payment grown by its own interest. That gives a formula, and logarithms let you solve for n. Compare two plans (different rates, fees or payments) and find when one becomes cheaper than the other. The maths starts gently, but the marks come from comparing plans and justifying the formula, not from an online calculator. At HL, derive it with the geometric series and take on something harder: changing rates, continuous compounding or a savings plan run backwards.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Use real offers

Two real plans from a bank or a dealer, with rates and fees as published.

→ Your data is real and the comparison matters; the assumptions (fees, rounding) are yours to justify.
Find the break-even

The month where the total cost of plan A equals plan B.

→ You solve an exponential equation with logarithms, or check numerically.
Change the compounding

Monthly, daily, continuous: compare the effective annual rates.

→ Links to the limit that defines e. At HL you can justify it with a series.
Change the payments

Extra money each month, or a lump sum at month 12.

→ Your formula has to change, and you test it against a month-by-month table.
Run it backwards

A regular saving plan that reaches a target.

→ The same series in the opposite direction; the annuity formula appears.
Let the rate change

The rate moves at month 24 or follows a published history.

→ The neat formula breaks into pieces: a recurrence or a spreadsheet helps, and HL maths justifies it.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • Names the plans being compared, with loan size, rate and term for each (real, or clearly assumed).
  • Names the quantity you compare: total interest, total cost, the balance at a given month, or the break-even month.
  • Says what your formula will be checked against (a month-by-month table, a bank's own schedule).
  • Is more than a calculator job: there is a formula to derive and a decision to justify.
  • States what you ignore: fees, inflation, tax, rounding to the cent.

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The model

Each month the balance grows by interest, then the payment comes off. Draw the same steps for your plans, with your own numbers.

B0B1B2B3B4B5B6B: what you still owe after n monthsinterest added on top......then the payment comes offhits zero when?do the same for plan 2: which hits zero first?

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; the plans you compare and why; what you expect before calculating.
Compound interest~2 pages
The formula derived from a geometric sequence; every symbol defined with units; how a yearly rate becomes a monthly one.
The repayment model~2.5 pages
The balance as a recurrence, then a closed formula. HL: derive it with the geometric series sum.
Checking the formula~1.5 pages
A month-by-month table for the first few months against the formula; agreement stated with numbers.
Comparing plans~3 pages
Both balances on one graph; total cost for each; the break-even found with logarithms and checked.
What if...~3 pages
One change (extra payments, a rate change, compounding frequency) and what it does to the answer.
Conclusion~1 page
Which plan, for whom, and when. Answer with numbers.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

mark the payoff month ✓

Made-up plans to show the form of a strong graph, not the answer: $10,000 borrowed at 0.5% interest per month. Yours will look different. example only

081624324048020004000600080001000012000Months since the loan began, nBalance owed B / $pay $300 a monthpay $200 a monthbalance hits zero here:solve with logarithmsILLUSTRATIVE · MADE-UP DATA
  • Axis units in dollars and months; say whether payments come at the start or end of each month.
  • Two plans on one set of axes: the difference is visible without reading a table.
  • The curve crossing zero is the payoff month: calculate it, don't read it off.
  • If you have a bank's schedule, plot it as points over your formula. Agreement is your evidence the formula is right.
  • Another graph to consider: total paid against time for both plans. Where the lines cross is the break-even.

How much interest does each plan cost?

rank by one number

Compare the plans on one number you can rank: total interest paid. Compute it for your own plans, and ask why the ordering comes out as it does.

$200 a month (58 months)1536$300 a month (37 months)967$400 a month (27 months)709lower payments mean more interest: why?Total interest paid on $10,000 at 0.5% per month, in dollarsILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

check against a table first

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictWrite down the compound interest idea in your own words. Predict which plan is cheaper overall and when, before any calculation.
  2. Collect real plansGet two or three real offers. Note rates, fees, terms and whether the rate is nominal or effective.
  3. Derive the formulaBuild the balance month by month, spot the geometric series and derive the closed formula. Define every symbol.
  4. Trial run (1 lesson)Make a spreadsheet table for six months and compare it with your formula. Fix any mismatch now: rounding, timing of payments, rate conversion.
  5. Solve for nUse logarithms to find when a balance reaches zero or when two costs are equal. Check the answer by substitution.
  6. Compare the plansPlot both on one graph, find the break-even and say what it means for a real borrower.
  7. Change one thingChoose your twist, adjust the formula and test it against a table.
  8. Test the assumptionsWhat changes if fees count, or the rate rises? Decide what to include, and say what you left out.
  9. Write, then cutWrite in layout order. Show one worked month, then a table, not twelve worked months.
  10. Check before your teacher's draftCheck notation (no calculator syntax), consistent rounding, graph titles with units, and reflection all the way through.

Maths you'll need

Traps to avoid in this Maths IA

  • Typing numbers into an online loan calculator and copying the answer: examiners want your formula and a check.
  • Mixing an annual rate with monthly payments without converting it.
  • A formula that is never checked against a month-by-month table.
  • Rounding to the cent in one place and not another, so the numbers disagree.
  • Ignoring fees or deposits that change which plan is cheaper.
  • Quoting the geometric series sum without saying why the series is geometric or what the ratio is.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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